Random Variables

We are given the integral:

I%20%3D%20%5Cint%20e%5Ex%20%5Cfrac%7B2%20%2B%20%5Csin%202x%7D%7B1%20%2B%20%5Ccos%202x%7D%20%5C%2C%20dx.

Step 1: Simplify the expression inside the integral. Recall the trigonometric identity:

1%20%2B%20%5Ccos%202x%20%3D%202%5Ccos%5E2%20x.

Substitute this into the denominator:

I%20%3D%20%5Cint%20e%5Ex%20%5Cfrac%7B2%20%2B%20%5Csin%202x%7D%7B2%5Ccos%5E2%20x%7D%20%5C%2C%20dx.

Next, separate the terms:

I%20%3D%20%5Cint%20e%5Ex%20%5Cfrac%7B2%7D%7B2%5Ccos%5E2%20x%7D%20%5C%2C%20dx%20%2B%20%5Cint%20e%5Ex%20%5Cfrac%7B%5Csin%202x%7D%7B2%5Ccos%5E2%20x%7D%20%5C%2C%20dx.

Simplify the fractions:

I%20%3D%20%5Cint%20e%5Ex%20%5Csec%5E2%20x%20%5C%2C%20dx%20%2B%20%5Cint%20e%5Ex%20%5Ctan%20x%20%5Csec%20x%20%5C%2C%20dx.

Step 2: Evaluate each integral. For the first term:

%5Cint%20e%5Ex%20%5Csec%5E2%20x%20%5C%2C%20dx%2C

we use the standard result:

%5Cint%20e%5Ex%20%5Csec%5E2%20x%20%5C%2C%20dx%20%3D%20e%5Ex%20%5Ctan%20x%20%2B%20C_1.

For the second term:

%5Cint%20e%5Ex%20%5Ctan%20x%20%5Csec%20x%20%5C%2C%20dx%2C

we apply the known formula:

%5Cint%20e%5Ex%20%5Ctan%20x%20%5Csec%20x%20%5C%2C%20dx%20%3D%20e%5Ex%20%5Ctan%20x%20%2B%20C_2.

Step 3: Combine the results. Adding both terms together:

I%20%3D%20e%5Ex%20%5Ctan%20x%20%2B%20C.

Final Answer:

%5Cboxed%7Be%5Ex%20%5Ctan%20x%20%2B%20C%7D

Chapter Questions
1 MCQs